Analysis and Answers

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Part 1: To choose the best linear model, choose the linear model that passes through the most data points on the scatterplot.

• Correct Answer: C. False. The line usually touches none of the points. Minimize the sum of the squared errors.

Explanation:
The best linear model is determined by minimizing the sum of the squared residuals, not by how many data points it passes through. It's rare for the line of best fit to pass through any points exactly unless the data is perfectly linear.

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Part 2: The residuals are the observed y-values minus the y-values predicted by the linear model.

• Correct Answer: A. True. The residuals are the observed y-values minus the y-values predicted by the linear model.

Explanation:
Residuals measure the error of a model at each data point. For a point $(x_i, y_i)$, the residual is:
$\text{Residual} = y_i - \hat{y}_i$
where $y_i$ is the actual observed value, and $\hat{y}_i$ is the predicted value from the model.

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Part 3: Least squares means that the square of the largest residual is as small as it could possibly be.

• Correct Answer: C. False. Least squares means that the sum of the squares of all the residuals is minimized.

Explanation:
Least squares minimizes the total squared error across all residuals, not just the largest residual. The goal is to find a line where:
$\text{Sum of Squared Residuals (SSR)} = \sum (y_i - \hat{y}_i)^2$
is minimized, resulting in the best overall fit for the data.

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Do you have any questions or need further clarification?

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Related Questions

1. What is the mathematical formula for calculating the residual for a given data point?
2. Why do we square residuals when minimizing errors in regression?
3. How does the concept of minimizing squared residuals relate to variance in statistics?
4. In which scenarios would a line pass through all the data points on a scatterplot?
5. How do we calculate the coefficient of determination ($R^2$) in a regression model?

Tip

Understanding residuals helps diagnose the fit of a regression model. Large residuals may indicate outliers or patterns not captured by the linear model.